Method of retrieving phase and magnitude of weak ultra-short optical pulses using a stronger unknown pulse

ABSTRACT

A method determines a complex electric field temporal profile of an optical pulse. The method includes providing a measured magnitude of the Fourier transform of a complex electric field temporal profile of a pulse sequence comprising the optical pulse and a dummy pulse. The method further includes providing an estimated phase term of the Fourier transform of the complex electric field temporal profile of the pulse sequence. The method further includes multiplying the measured magnitude and the estimated phase term to generate an estimated Fourier transform of the complex electric field temporal profile of the pulse sequence. The method further includes calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a function of time. The method further includes calculating an estimated complex electric field temporal profile of the pulse sequence by applying at least one constraint to the inverse Fourier transform.

CLAIM OF PRIORITY

This application claims the benefit of U.S. Provisional Application No.60/662,601, filed Mar. 17, 2005, which is incorporated in its entiretyby reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to apparatus and methods ofdetermining the phase and magnitude of optical pulses.

2. Description of the Related Art

Ultra-short optical pulses with sub-picosecond time scales play a keyrole in many important applications such as medical imaging, surgery,micro-machining, optical communication, and 3D optical waveguidefabrication. (See, e.g., Jean-Claude Diels and Wolfgang Rudolph,“Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques andApplications on a Femtosecond Time Scale,” Elsevier, Academic Press,London (1996); M. R. Hee et al., “Femtosecond transilluminationtomography in thick tissue,” Opt. Lett., Vol. 18, pp. 1107-1109 (1993);X. Liu et al., “Laser ablation and micromachining with ultrashort laserpulses,” IEEE J. Quant. Electr., Vol. 33, pp. 1706-1716, (1997); K. M.Davis et al., “Writing waveguides in glass with a femtosecond laser,”Opt. Lett., Vol. 21, pp. 1729-1731 (1996); A. M. Weiner et al.,“Encoding and decoding of femtosecond pulses,” Opt. Lett., Vol. 13, pp.300-302 (1988).)

In many of these applications, knowledge of the temporal profile of theoptical pulse (both its phase and magnitude) is important. Over the lastdecade, many techniques have been developed to characterize ultra-shortoptical pulses. (See, e.g., K. L. Sala et al., “CW autocorrelationmeasurements of picosecond laser pulses,” IEEE J. Quant. Electr., Vol.QE-16, pp. 990-996 (1980); J. L. A. Chilla and O. E. Martinez, “Directdetermination of the amplitude and the phase of femtosecond lightpulses,” Opt. Lett., Vol. 16, pp. 39-41 (1991); J. Peatross and A.Rundquist, “Temporal decorrelation of short laser pulses,” J. Opt. Soc.Am. B, Vol. 15, 216-222 (1998); J. Chung and A. M. Weiner, “Ambiguity ofultrashort pulse shapes retrieved from the intensity autocorrelation andthe power spectrum,” IEEE J. Select. Quantum Electron. pp. 656-666(2001).)

These techniques can generally be divided into two categories: nonlinearand linear. Nonlinear techniques typically use a thin nonlinear crystal.The well-known nonlinear techniques include frequency-resolved opticalgating (FROG) (see, e.g., R. Trebino and D. J. Kane, “Using phaseretrieval to measure the intensity and phase of ultrashort pulses:frequency-resolved optical gating,” J. Op. Soc. Am. A, Vol. 10, pp.1101-1111 (1993)), spectral phase interferometry for directelectric-field reconstruction (SPIDER) (see, e.g., C. Iaconis and I. A.Walmsley, “Spectral phase interferometry for direct electric-fieldreconstruction of ultrashort optical pulses,” Opt. Lett., Vol. 23, pp.792-794 (1998)), spectrally resolved cross-correlation (XFROG) (see,e.g., S. Linden et al., “XFROG—A new method for amplitude and phasecharacterization of weak ultrashort pulses,” Phys. Stat. Sol. (B), Vol.206, pp. 119-124 (1998)), and phase and intensity from cross-correlationand spectrum only (PICASO) (see, e.g., J. W. Nicholson et al.,“Full-field characterization of femtosecond pulses by spectrum andcross-correlation measurements,” Opt. Lett., Vol. 24, pp. 1774-1776(1999)). Because the nonlinear process is generally weak, thesetechniques tend to require high peak powers and are generally notsuitable for characterizing weak optical pulses.

Linear techniques were conceived in part to eliminate this powerlimitation. One exemplary linear technique is spectral interferometry(SI), which uses a linear detection system, such as an optical spectrumanalyzer (OSA), to record in the frequency domain the interferencebetween the sample pulse to be characterized and a reference pulse.(See, e.g., D. E. Tokunaga et al., “Femtosecond continuum interferometerfor transient phase and transmission spectroscopy,” J. Opt. Soc. Am. B,Vol. 13, pp. 496-513 (1996); D. Meshulach et al., “Real-timespatial-spectral interference measurements of ultrashort opticalpulses,” J. Opt. Soc. Am. B, Vol. 14, pp. 2095-2098 (1997).) Temporalanalysis by dispersing a pair of light electric fields (TADPOLE) (see,e.g., D. N. Fittinghoff et al., “Measurement of the intensity and phaseof ultraweak, ultrashort laser pulses,”. Opt. Lett., Vol. 21, pp.884-886 (1996)) is a popular SI technique. Using the TADPOLE technique,the reference pulse is first fully characterized using a FROG set-up,then an OSA is used to measure the power spectra of the sample pulse andof a pulse sequence formed by delaying the reference pulse with respectto the sample pulse. These three measurements enable the recovery of thefull complex electric field of the sample pulse, even if this pulse isvery weak. Note that SI-based techniques utilize a fully-characterizedreference pulse.

SUMMARY OF THE INVENTION

In certain embodiments, a method determines a complex electric fieldtemporal profile of an optical pulse. The method comprises providing ameasured magnitude of the Fourier transform of a complex electric fieldtemporal profile of a pulse sequence comprising the optical pulse and adummy pulse. The method further comprises providing an estimated phaseterm of the Fourier transform of the complex electric field temporalprofile of the pulse sequence. The method further comprises multiplyingthe measured magnitude and the estimated phase term to generate anestimated Fourier transform of the complex electric field temporalprofile of the pulse sequence. The method further comprises calculatingan inverse Fourier transform of the estimated Fourier transform, whereinthe inverse Fourier transform is a function of time. The method furthercomprises calculating an estimated complex electric field temporalprofile of the pulse sequence by applying at least one constraint to theinverse Fourier transform.

In certain embodiments, a computer system comprises means for estimatingan estimated phase term of a Fourier transform of a complex electricfield temporal profile of a pulse sequence comprising an optical pulseand a dummy pulse. The computer system further comprises means formultiplying a measured magnitude of the Fourier transform of the complexelectric field temporal profile of the pulse sequence and the estimatedphase term to generate an estimated Fourier transform of the complexelectric field temporal profile of the pulse sequence. The computersystem further comprises means for calculating an inverse Fouriertransform of the estimated Fourier transform, wherein the inverseFourier transform is a function of time. The computer system furthercomprises means for calculating an estimated complex electric fieldtemporal profile by applying at least one constraint to the inverseFourier transform.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a plot of the complex electric field temporal profile(magnitude and phase shown by solid and dashed lines, respectively) of apulse sequence formed by delaying a sample pulse in time with respect toa stronger dummy pulse.

FIG. 1B is a plot of the power spectrum (i.e., square of the Fouriertransform magnitude) of the complex electric field temporal profile ofFIG. 1A.

FIG. 2 is a plot of the original complex electric field temporal profileof FIG. 1A (solid lines) and a recovered complex electric field temporalprofile (dashed lines) recovered using an exemplary embodiment from thepower spectrum of FIG. 1B.

FIG. 3 is a flow diagram of an exemplary iterative error-reductionmethod compatible with certain embodiments described herein.

FIG. 4A is a plot of an exemplary complex electric field temporalprofile of a pulse sequence, with the magnitude shown by the solid lineand the phase shown by the dashed line.

FIG. 4B is a pole-zero plot of the z-transform of the pulse sequence ofFIG. 4A with zeros shown as circles.

FIGS. 5A and 5B are plots of the magnitude and phase, respectively, ofthe original complex electric field temporal profile (solid lines) andthe recovered complex electric field temporal profile (dashed lines)recovered by applying an exemplary embodiment to the Fourier transformmagnitude of FIG. 4A.

FIG. 6 is a pole-zero plot of the z-transform of the pulse sequence of anew pulse sequence formed in which the first waveform peak of FIG. 4Anear t=0 is increased from approximately 0.4 to approximately 50, withzeros shown as circles.

FIGS. 7A and 7B are plots of the magnitude and phase, respectively, ofthe original complex electric field temporal profile (solid lines) andthe recovered complex electric field temporal profile (dashed lines)recovered by applying an exemplary embodiment to the Fourier transformmagnitude of a new pulse sequence having a large waveform peak at t=0.

FIG. 8A is a flow diagram of a method of determining the complexelectric field temporal profile of a sample optical pulse.

FIG. 8B schematically illustrates a configuration compatible withcertain embodiments described herein.

FIG. 9A illustrates an exemplary sample pulse with the magnitude shownas a solid line and the phase shown as a dashed line.

FIG. 9B illustrates an exemplary dummy pulse with the magnitude shown asa solid line and the phase shown as a dashed line.

FIG. 10A illustrates the calculated magnitude (solid line) and phase(dashed line) of the Fourier transform spectrum of the sample pulse ofFIG. 9A.

FIG. 10B illustrates the calculated magnitude (solid line) and phase(dashed line) of the Fourier transform spectrum of the dummy pulse ofFIG. 9B.

FIG. 11 is a plot of the complex electric field temporal profiles of theoriginal dummy pulse of FIG. 9A (solid line) and the recovered dummypulse (dashed line).

FIG. 12 is a log-log plot of the error in the recovered electric fieldmagnitude and phase as functions of the ratio of the peak magnitude ofthe dummy pulse to that of the sample pulse.

FIG. 13 is a plot of a simulated noisy power spectrum (solid line) and asimulated noise-free power spectrum (dashed line).

FIGS. 14A and 14B are plots of the magnitude and phase, respectively, ofthe original complex electric field temporal profile (solid lines) andthe recovered complex electric field temporal profile (dashed lines)recovered by applying an exemplary embodiment to the noisy powerspectrum of FIG. 13.

FIG. 15A illustrates an exemplary sample pulse with the magnitude shownas a solid line and the phase shown as a dashed line.

FIG. 15B illustrates an exemplary dummy pulse which is temporallynarrower than the sample pulse of FIG. 15A with the magnitude shown as asolid line and the phase shown as a dashed line.

FIG. 16A illustrates the calculated magnitude (solid line) and phase(dashed line) of the Fourier transform spectrum of the sample pulse ofFIG. 15A.

FIG. 16B illustrates the calculated magnitude (solid line) and phase(dashed line) of the Fourier transform spectrum of the dummy pulse ofFIG. 15B.

FIG. 17A is a plot of the complex electric field temporal profile(magnitude and phase shown by solid and dashed lines, respectively) of apulse sequence formed by delaying a sample pulse in time with respect toa stronger dummy pulse.

FIG. 17B is a plot of the power spectrum (i.e., square of the Fouriertransform magnitude) of the complex electric field temporal profile ofFIG. 17A.

FIG. 18 is a plot of the original complex electric field temporalprofile of FIG. 17A (solid lines) and a recovered complex electric fieldtemporal profile (dashed lines) recovered using an exemplary embodimentfrom the power spectrum of FIG. 17B.

FIG. 19A is a plot of the complex electric field temporal profile(magnitude and phase shown by solid and dashed lines, respectively) of apulse sequence formed by a dummy pulse and two sample pulses.

FIG. 19B is a plot of the power spectrum (i.e., square of the Fouriertransform magnitude) of the complex electric field temporal profile ofFIG. 19A.

FIG. 20 is a plot of the original complex electric field temporalprofile of FIG. 19A (solid lines) and a recovered complex electric fieldtemporal profile (dashed lines) recovered using an exemplary embodimentfrom the power spectrum of FIG. 19B.

FIG. 21 schematically illustrates an example measurement configurationcompatible with certain embodiments described herein.

FIG. 22 is a graph of an example power spectrum of the pulse sequenceformed by delaying the dummy pulse with respect to the sample pulsemeasured using the measurement configuration of FIG. 21.

FIGS. 23A and 23B are graphs of the resulting intensity and phase,respectively, of the recovery after applying the technique disclosedherein to the square-root of the measured power spectrum shown in FIG.22, as compared to the results of the recovery obtained using both theFROG technique and the TADPOLE technique.

FIGS. 24A and 24B show recorded charge-coupled-device (CCD) images atthe optical spectrum analyzer for two successive measurements withslightly different delay values between the sample pulse and the dummypulse.

FIGS. 25A and 25B are plots of the measured power spectrum correspondingto the input pulse sequence obtained from FIGS. 24A and 24B,respectively, by adding the recorded spatial Fourier transform magnitudespectra along the vertical axis.

FIGS. 26A and 26B show the intensity and phase, respectively, of theelectric field of the sample pulse recovered by applying the techniquedisclosed herein to the square-root of the measured power spectra ofFIGS. 25A (“SIMBA measurement #1”) and 25B (“SIMBA measurement #2”), ascompared to the results of the FROG technique applied to the same samplepulse.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Certain embodiments described herein are useful in computer-implementedanalyses of the temporal waveforms of optical pulses. Thegeneral-purpose computers used for such analyses can take a wide varietyof forms, including network servers, workstations, personal computers,mainframe computers and the like. The code which configures the computerto perform such analyses is typically provided to the user on acomputer-readable medium, such as a CD-ROM. The code may also bedownloaded by a user from a network server which is part of a local-areanetwork (LAN) or a wide-area network (WAN), such as the Internet.

The general-purpose computer running the software will typically includeone or more input devices, such as a mouse, trackball, touchpad, and/orkeyboard, a display, and computer-readable memory media, such asrandom-access memory (RAM) integrated circuits and a hard-disk drive. Itwill be appreciated that one or more portions, or all of the code may beremote from the user and, for example, resident on a network resource,such as a LAN server, Internet server, network storage device, etc. Intypical embodiments, the software receives as an input a variety ofinformation concerning the material (e.g., structural information,dimensions, previously-measured magnitudes of reflection or transmissionspectra).

Certain embodiments described herein provide a novel linear method,referred to herein as “SIMBA”, which uses a single optical spectrumanalyzer (“OSA”) measurement to recover the phase and magnitude of thecomplex electric field of weak ultra-short optical pulses. As usedherein, the term “SIMBA” refers to either “spectral interferometry usingminimum-phase-based algorithms” or “spectral interferometry usingmaximum-phase-based algorithms. Certain embodiments described herein areamong the simplest and fastest of all the methods reported to date tomeasure the temporal shape of ultra-short optical pulses. Certainembodiments described herein are broadly applicable since the conditionsof such embodiments are relatively lax as compared to otherpreviously-used methods. In certain embodiments, SIMBA involves using anOSA to measure the power spectrum of a sequence of two pulses: areference or dummy pulse combined with a sample pulse of weakermagnitude. Such a pulse sequence in which a large dummy pulse isfollowed by a weaker sample pulse approximates a minimum-phase functionin certain embodiments. Such a pulse sequence in which a weaker samplepulse is followed by a large dummy pulse approximates a maximum-phasefunction in certain embodiments. In certain embodiments, the temporalprofile of the sample pulse is recoverable using only the magnitude ofthe Fourier transform (e.g., the square root of the measured powerspectrum) of the pulse sequence. As described below, this recovery ofthe temporal profile of the sample pulse can be carried out numericallywith a simple iterative method that takes only seconds on a 500-MHzcomputer using MATLAB 5. With a faster computer and programming tool,this method has the capability to provide real-time dynamic measurementsof laser pulse profiles.

Certain embodiments described herein provide various advantages overexisting pulse-profile characterization methods which make SIMBA anexcellent candidate for accurate, real-time characterization ofultrashort laser pulses. In certain embodiments, the temporal profile ofthe dummy pulse does not need to be known. Such embodiments canadvantageously provide a significant time saving over other SItechniques which require first characterizing the dummy pulse. Certainembodiments advantageously work with weak sample pulses, unlikenonlinear techniques, as well as with strong sample pulses. Themeasurement configuration of certain embodiments described herein isadvantageously simple by utilizing an OSA and not containing any movingparts. Certain embodiments advantageously utilize a single measurement.As compared to previously-known techniques (e.g., PICASO or TADPOLEwhich require 2 and 3 simultaneous measurements, respectively), certainsuch embodiments advantageously provide a fast determination of thetemporal profile of the optical pulse. Certain embodiments are fastenough to allow real-time characterization of an optical pulse. Unlikecertain previously-known techniques (e.g., second-harmonic FROG),certain embodiments advantageously can differentiate an ultrashortoptical pulse from its time-reversed replica. Unlike many other SI-basedtechniques, certain embodiments described herein do not have any minimumconstraint for the time delay between the dummy pulse and the samplepulse. Certain embodiments described herein can advantageously be usedto simultaneously characterize a sequence of different sample pulseswith a single measurement.

Certain embodiments described herein utilize a property of minimum-phasefunctions (MPFs) that allows the phase of the Fourier transform (FT) ofthe minimum-phase function to be extracted from its FT magnitude alone,either analytically or iteratively. (See, e.g., V. Oppenheim and R. W.Schafer, Digital Signal Processing, Prentice Hall, 2002, Chap. 7; T. F.Quatieri, Jr., and A. V. Oppenheim, “Iterative techniques for minimumphase signal reconstruction from phase or magnitude,” IEEE Trans.Acoust., Speech, Signal Processing, Vol. 29, pp. 1187-1193 (1981); M.Hayes et al., “Signal reconstruction from phase or magnitude,” IEEETrans. Acoust., Speech, Signal Processing, Vol. 28, pp. 672-680 (1980).)Similarly, certain embodiments described herein utilize the sameproperty of maximum-phase functions. While certain embodiments aredescribed below by referring to MPFs, certain other embodimentssimilarly utilize maximum-phase functions.

FIGS. 1A, 1B, and 2 illustrate the performance of certain embodimentsdescribed herein using numerical simulations of an arbitrary samplepulse. FIG. 1A illustrates the magnitude and phase of the electric fieldof a sequence of two pulses, namely a strong reference or dummy pulsefollowed by a sample pulse to be characterized. The sample pulse of FIG.1A has an arbitrarily chosen temporal profile. A measured optical powerspectrum of this pulse sequence (e.g., by sending the pulse sequenceinto an OSA, including a noise contribution, discussed more fully below)was simulated numerically, and is illustrated by FIG. 1B. The magnitudeand phase of the weak sample pulse recovered by applying SIMBA to theoptical power spectrum of FIG. 1B, without any other knowledge of thepulse sequence, is plotted in FIG. 2. The excellent accuracy of therecovery of the sample pulse temporal profile can be achieved with anysample pulse with properly-chosen properties (bandwidth and magnitude)of the dummy pulse and these properly-chosen properties are notexcessively restrictive.

It is generally not possible to fully recover a one-dimensional functionfrom the knowledge of its FT magnitude alone. However, there arefamilies of functions which are exceptions to this rule for which the FTphase can be recovered from the FT magnitude alone, and visa versa. Oneexemplary such family is the family of minimum-phase functions (MPFs).An MPF is characterized by having a z-transform with all its poles andzeros either on or inside the unit circle. As a result of this property,the FT phase and the logarithm of the FT magnitude of an MPF are theHilbert transforms of one another. Consequently, the FT phase of an MPFcan be calculated from its FT magnitude, and an MPF can be reconstructedfrom its FT magnitude alone.

This reconstruction can be done by first taking the Hilbert transform ofthe logarithm of the function's FT magnitude (e.g., the logarithmicHilbert transform of the function's FT magnitude) to obtain the FTphase, and then inverting the full complex FT. However, this directapproach can have difficulties in its implementation, such as phaseunwrapping.

A second approach for the reconstruction is to use an iterativeerror-reduction method. Examples of iterative error-reduction methodsinclude, but are not limited to, those described by J. R. Fienup,“Reconstruction of an object from the modulus of its Fourier transform,”Opt. Lett., Vol. 3, pp. 27-29 (1978) or R. W. Gerchberg and W. O.Saxton, “Practical algorithm for the determination of phase from imageand diffraction plane pictures,” Optik, Vol. 35, pp. 237-246 (1972).

FIG. 3 is a flow diagram of an exemplary iterative error-reductionmethod 100 compatible with certain embodiments described herein. Thisiterative error-reduction 100 involves using a known (e.g., measured)Fourier transform magnitude spectrum of an unknown function e(t),together with known properties of this function (e.g., that it is a realfunction or a causal function), to correct an initial guess of e(t). Incertain embodiments, this correction is done iteratively. In certainembodiments, the unknown function e(t) comprises the complex electricfield temporal profile of a pulse sequence comprising a sample pulse anda dummy pulse, as described more fully below.

Given a complex MPF, e(t), the only quantity that is fed into the method100 is the FT magnitude spectrum of e(t), i.e., |E_(M)(f)|, where thesubscript M denotes that this spectrum is a measured quantity, as shownby the operational block 110. In certain embodiments, providing themeasured FT magnitude spectrum comprises measuring a power spectrum of apulse sequence comprising a sample optical pulse and a dummy pulse andcalculating the square root of the measured power spectrum to yield themeasured FT magnitude spectrum. In certain other embodiments, providingthe measured FT magnitude spectrum comprises providing apreviously-measured power spectrum of a pulse sequence comprising asample optical pulse and a dummy pulse and calculating the square rootof the previously-measured power spectrum.

Since the FT phase is missing, an initial guess, φ₀(f), for this phaseis provided in the operational block 120. In certain embodiments, thisguess does not significantly affect the accuracy of the result of theconvergence of the method 100. For this reason, φ₀(f) can beconveniently chosen to equal zero (e.g., φ₀(f)=0) or some other real orcomplex constant (e.g., π, π/2). In certain embodiments, the initialguess for the phase can be a previously-stored function φ₀(f) retrievedfrom the computer system. In certain embodiments, the initial guess forthe phase can be a phase calculated from a previous optical pulse. Incertain embodiments, the initial guess for the phase can be calculatedfrom the measured magnitude using a logarithmic Hilbert transform.

In certain embodiments, the inverse Fourier transform (IFT) of|E_(M)|·exp(jφ₀) is then computed numerically, as shown by theoperational block 130, yielding a function e′(t). In certainembodiments, the operational block 140 comprises applying at least oneconstraint to the estimated function e′(t). For example, in certainembodiments in which the pulse sequence approximates a minimum-phasefunction (MPF) (e.g., the dummy pulse precedes the sample pulse), sinceMPFs are causal, only the t≧0 portion of e′(t) is retained (e.g., thecausality condition), and all values of e′(t) for t<0 are set to zero,thereby producing a new function e₁(t). In certain embodiments in whichthe pulse sequence approximates a maximum-phase function (e.g., thesample pulse precedes the dummy pulse), since maximum-phase functionsare anti-causal, only the t≦0 portion of e′(t) is retained (e.g., theanti-causality condition), and all values of e′(t) for t>0 are set tozero, thereby producing a new function e₁(t). The recovered dummy pulsein such embodiments is on the negative time axis close to the origin andthe sample pulse is recovered in the deeper part of the negative timeaxis.

In certain embodiments in which e(t) is known to be limited in time(e.g., to be less than 100 femtoseconds long), the operational block 140can also include inserting zeros for times greater than this limit(e.g., t>100 femtoseconds) to produce the function e₁(t), therebyadvantageously speeding up convergence of the method 100. In certainembodiments in which the maximum peak power of the laser pulses arepredetermined or known, the magnitudes of the intermediate functions canbe constrained to be below or equal to the maximum peak power. Incertain embodiments, the new function e₁(t) provided by the operationalblock 140 serves as a first estimate of the complex MPF.

In certain embodiments, the FT of e₁(t) is calculated in the operationalblock 150, thereby providing a new phase φ₁(f) and a new magnitude|E₁(f)| for the FT of e(t). In certain embodiments, the magnitude of thecalculated FT spectrum |E₁(f)| is replaced by the measured magnitude|E_(M)(f)|, as shown by the arrow 160. In certain embodiments, the loopis then repeated using |E_(M)(f)| and φ₁(f) as the new input function inthe operational block 130, which provides a second function e₂(t). Incertain embodiments, only a single iteration is used, while in otherembodiments, this loop is repeated until convergence is achieved. Incertain embodiments, convergence is defined to be achieved when thedifference between consecutive estimates of the function∫|e_(n)(t)−e_(n-1)(t)|²dt/∫|²dt is less than a predetermined value, forexample 0.1%. In certain embodiments, less than 100 iterations areadequate for achieving convergence, taking a few seconds to computeusing MATLAB 5 on a 500 MHz computer with 2¹⁴ data points. In certainembodiments, applying the constraint in the operational block 140advantageously reduces the number of iterations which achieveconvergence.

In certain other embodiments, the loop is repeated a predeterminednumber of times (e.g., 100). In certain embodiments, the predeterminednumber of times is selected to be sufficiently large so that the methodachieves, or is close to achieving, convergence. In certain embodiments,at the end of the n-th iteration, e_(n)(t) is the recovered complex MPF.

Empirical results indicate that such iterative error-reduction methodsconverge to the minimum-phase function corresponding to a given FTmagnitude. (See, e.g., T. F. Quatieri, Jr., and A. V. Oppenheim,“Iterative techniques for minimum phase signal reconstruction from phaseor magnitude,” IEEE Trans. Acoust., Speech, Signal Processing, Vol. 29,pp. 1187-1193 (1981); A. Ozcan et al., “Iterative processing ofsecond-order optical nonlinearity depth profiles,” Opt. Express, Vol.12, pp. 3367-3376 (2004); A. Ozcan et al., “Group delay recovery usingiterative processing of amplitude of transmission spectra of fibre Bragggratings,” Electron. Lett., Vol. 40, pp. 1104-1106 (2004).) In otherwords, for the infinite family of FT phase functions that can beassociated with a known (e.g., measured) FT magnitude, certainembodiments described herein converge to the one and only one FT phasefunction that has the minimum phase. Since this solution is unique, ifit is known a priori that the profile to be reconstructed is an MPF (orthat the profile approximates an MPF), then the solution provided by theerror-reduction method is the correct profile. Similarly, if it is knowna priori that the profile to be reconstructed is a maximum-phasefunction (or that the profile approximates a maximum-phase function),then the solution provided by the error-reduction method is the correctprofile.

To understand intuitively which physical functions are likely to beminimum-phase functions, an MPF is denoted by e_(min)(n), where n is aninteger that corresponds to sampled values of the function variable(e.g., relative time for the temporal waveform of ultra-short pulses).Because all physical MPFs are causal, e_(min)(n) equals to zero fortimes less than zero (e.g., for n<0).

The energy of an MPF, defined as$\sum\limits_{n = 0}^{m - 1}{{e_{\min}(n)}}^{2}$for m samples of the function e_(min)(n), satisfies the inequality:${\sum\limits_{n = 0}^{m - 1}{{e_{\min}(n)}}^{2}} \geq {\sum\limits_{n = 0}^{m - 1}{{e(n)}}^{2}}$all possible values of m>0. In this inequality, e(n) represents any ofthe functions that have the same FT magnitude as e_(min)(n). Thisproperty suggests that most of the energy of e_(min)(n) is concentratedaround n=0. Stated differently, any profile with a dominant peak aroundn=0 (e.g., close to the origin) will be either a minimum-phase functionor close to one, and thus the profile will work extremely well with theiterative error-reduction method 100 outlined by FIG. 3. Although theremay be other types of MPFs besides functions with a dominant peak, thisclass of MPFs can be used as described herein because they arestraightforward to construct with optical pulses and because they yieldexceedingly good results.

To further illustrate the utility of a dominant peak close to theorigin, FIG. 4A illustrates an exemplary complex electric field temporalprofile of an optical pulse. As illustrated by FIG. 4A, the magnitude ofthis causal function has four peaks. One peak close to the origin (e.g.,t=0) has a magnitude of approximately 0.4, and a dominant peak atapproximately t=0.1 has a magnitude of approximately 1.0. Since thedominant peak is not the peak closest to the origin, this function isnot an MPF. This result can be verified by the pole-zero plot of thepulse sequence's z-transform, as shown in FIG. 4B. Many of the zeros ofthe z-transform of the pulse sequence lie outside the unit circle,therefore the pulse sequence of FIG. 4A is not an MPF, and the phase andmagnitude of the pulse sequence's FT cannot be accurately related by thelogarithmic Hilbert transform or by iterative error-reduction methods.

The dashed curves of FIGS. 5A and 5B illustrate the magnitude and phase,respectively, of the complex electric field temporal profile recoveredby applying the iterative method 100 of FIG. 3 to the FT magnitude ofFIG. 4A. Both the recovered magnitude and phase functions aresubstantially different from the original functions, shown by the solidcurves of FIGS. 5A and 5B (which are the same as those in FIG. 4A). Asshown in FIG. 5A, the magnitude of the temporal profile recovered by theerror-reduction method of FIG. 3 exhibits a large peak near t=0. Thereason for this peak is that the method 100 converges to theminimum-phase function associated with the original FT magnitudespectrum, which must have a dominant peak near the origin.

To illustrate aspects of certain embodiments of the method describedherein, the magnitude of the first waveform peak near t=0 is increasedfrom approximately 0.4, as shown in FIG. 4A, to a large value (e.g.,approximately 50), so that this peak becomes the dominant peak. Thephase of the waveform remains unchanged from that shown in FIG. 4A. Thepole-zero plot of the new pulse sequence is illustrated by FIG. 6, andshows that almost all of the zeros of its z-transform are now either inor very close to the unit circle. Increasing the magnitude of the firstpeak pushes all the zeros of the temporal profile's z-transform towardsthe unit circle, thereby making the new pulse sequence closer to a trueMPF. Mathematically, by increasing the magnitude of the first peak, theinequality${{\sum\limits_{n = 0}^{m - 1}{{e_{\min}(n)}}^{2}} \geq {\sum\limits_{n = 0}^{m - 1}{{e(n)}}^{2}}},$which is another definition of an MPF, becomes easier to satisfy for allpossible values of m>0. Since this new pulse sequence is very close toan MPF, the phase and magnitude of its FT are accurately related by thelogarithmic Hilbert transform or by iterative methods. FIGS. 7A and 7Billustrate the magnitude and phase, respectively, of the originalfunction (solid line) and the complex function recovered (dashed line)by applying the iterative method of FIG. 3 to the FT magnitude of thenew pulse sequence. As shown by FIGS. 7A and 7B, the agreement betweenthe original temporal profile and the recovered temporal profile isexcellent.

FIG. 8A is a flow diagram of a method 200 of determining the complexelectric field temporal profile of a sample optical pulse 202. FIG. 8Bschematically illustrates a measurement configuration compatible withcertain embodiments described herein. In an operational block 210, thesample optical pulse 202 is provided. In certain embodiments, the sampleoptical pulse 202 is an ultra-short optical pulse having a pulse widthin a range between approximately 10 femtoseconds and approximately 1picosecond. Other pulse widths are also compatible with certainembodiments described herein.

In an operational block 220, a pulse sequence 222 is formed by combininga strong dummy pulse 224 and the sample optical pulse 202 with a timeperiod between the dummy pulse 224 and the sample optical pulse 202. Incertain embodiments, the dummy pulse 224 precedes the sample pulse 202(e.g., the dummy pulse 224 enters the spectrum analyzer 232 before thesample pulse 202 does) such that the pulse sequence 222 approximates aminimum-phase function. In certain other embodiments, such as thatschematically illustrated by FIG. 8B, the sample pulse 202 precedes thedummy pulse 224 (e.g., the sample pulse 202 enters the spectrum analyzer232 a time period π before the dummy pulse 224 does). In certain suchembodiments, the pulse sequence 222 approximates a maximum-phasefunction.

In an operational block 230, a square of the FT magnitude of the pulsesequence is measured. In certain embodiments, this measurement isperformed by sending the pulse sequence 222 into an OSA 232. The OSA 232of certain embodiments comprises a computer which determines the complexelectric field temporal profile of the sample pulse 202. The OSA 232 ofcertain other embodiments is coupled to a computer 240 which determinesthe complex electric field temporal profile of the sample pulse 202. Asdiscussed above, by adding a strong dummy pulse 224 to the sample pulse202, the complex temporal profile of the pulse sequence 222 approximatesan MPF or a maximum-phase function. The complex temporal profile of thesample pulse 202 is then recoverable from the measured spectrum using anerror-reduction method such as the one shown in FIG. 3.

FIG. 9A illustrates an exemplary sample pulse 202 used to demonstrate anexemplary embodiment of the SIMBA method step-by-step. The solid curveof FIG. 9A is the magnitude of the temporal electric field profile ofthe sample pulse 202, and the dashed curve is its phase. This samplepulse 202 was selected arbitrarily, except that its FT was selected tohave a strong second-order phase, as shown in FIG. 10A. In certain suchembodiments, the strong second-order phase makes it more difficult toretrieve the complex temporal profile of the sample pulse 202. The phaseand magnitude of the sample pulse 202 were also selected such that theFT magnitude spectrum, calculated numerically and shown in FIG. 10A,resembles the spectrum of typical optical pulses expected to bedetermined using embodiments described herein.

FIG. 9B shows the magnitude and phase of the reference or dummy pulse224. The calculated FT spectrum of this dummy pulse 224 is plotted inFIG. 10B. The shape of the dummy pulse 224 of FIG. 10B was selectedarbitrarily, with a large amount of third-order phase in its FT. Thisfeature of the dummy pulse 224 is responsible for the oscillations inthe tail of the pulse magnitude illustrated by FIG. 9B. As disclosedabove, the peak magnitude of the dummy pulse electric field was taken tobe much larger than that of the sample pulse 202. For the sample pulse202 and the dummy pulse 224 of FIGS. 9A and 9B, the peak magnitude ofthe dummy pulse 224 is larger than that of the sample pulse 202 by afactor of approximately 40, corresponding to a factor of approximately1600 in power.

In certain embodiments, the sample pulse 202 is delayed in time by τwith respect to the dummy pulse 224 to form the pulse sequence 222. Asshown in FIG. 1A, this pulse sequence 222 approximates an MPF. Incertain embodiments, the time delay τ is sufficiently large so that thesample pulse 202 and the dummy pulse 224 do not overlap. This conditionis not as strict as conditions required by other SI techniques whichrequire larger minimum delays to avoid aliasing in the inverse FTdomain, which would make the recovery impossible. In certain embodimentsdescribed herein, aliasing can be present because it does not affect therecovery process. The delay τ of certain embodiments is sufficientlysmall so as to avoid high-frequency oscillations in the measuredspectrum of the pulse sequence 222, which would require a higherresolution OSA. Sending the pulse sequence 222 into an OSA yields thesquare of the FT magnitude of the complex electric field temporalprofile, as shown in FIG. 1B. The oscillations near the center of thepower spectrum of FIG. 1B are due to the interference between theelectric fields of the sample pulse 202 and the dummy pulse 224.

In certain embodiments, the power spectrum, such as illustrated by FIG.1B, contains all the necessary information to recover the complexelectric field of the sample pulse 202. In certain embodiments, thisrecovery is achieved by applying the method 100 shown in FIG. 3 to thesquare root |E_(M)(f)| of the power spectrum of FIG. 1B. The magnitudeand phase of the electric field of the sample pulse 202 obtained by thismethod 100 are plotted as dashed curves in FIG. 2. Comparison to themagnitude and phase of the original sample pulse 202, reproduced fromFIG. 9A and included in FIG. 2 as solid curves, shows that both themagnitude and phase components of the recovered sample pulse arevirtually indistinguishable from the components of the original samplepulse 202.

Certain embodiments described herein advantageously recover the complexelectric field of an optical pulse from a single FT magnitudemeasurement, without any additional information about the reference ordummy pulse 224. The result of FIG. 2 utilized a computation time ofonly a few seconds using MATLAB 5 on a 500-MHz computer. The number ofdata points used to simulate the measured FT magnitude, shown in FIG.1B, was limited to approximately 1500. To increase the resolution andspeed of the processing (which involves fast FT routines), the totalnumber of points was increased to approximately 2¹⁴ by zero padding.Real-time characterization of ultra-short optical pulses usingembodiments described herein is possible by using a much fasterprogramming environment such as C++. In certain embodiments, thisrecovery could be done using the analytical logarithmic Hilberttransform instead of an iterative approach, although the results of sucha recovery could be more susceptible to noise.

In certain embodiments in which the arrival times of the input pulsesare not recorded, the recovered temporal profile exhibits a time shiftcompared to the actual temporal profile. FIG. 2 does not show this timeshift because, for comparison purposes, the recovered profile wasshifted to match the original profile. This time shift is common to manyexisting recovery techniques, and it is totally inconsequential.

The recovered phase shown in FIG. 2 has a slight, inconsequentialdeviation, especially towards the edges of FIG. 2. This deviation isprimarily due to the intensity of the optical pulse is very weak atthose times, which makes the phase recovery less accurate. In thelimiting case in which the pulse intensity goes to zero at these times,the recovery of the exact phase becomes almost impossible. However,since the phase of a zero field is meaningless, this limitation isinconsequential. This same phenomena is also present in other existingtechniques (e.g., FROG, TADPOLE, and SPIDER).

In certain embodiments, the recovery of the temporal profile of theoptical pulse after applying time-reversal to at least one of thecomponents of the pulse sequence (e.g., the sample pulse 202, the dummypulse 224, or both the sample pulse 202 and the dummy pulse 224) is asaccurate as the recovery of the temporal profile of the optical pulsewithout time-reversal being applied. This result indicates that certainembodiments described herein can conveniently differentiate between apulse and its time-reversed version. This result is a significantimprovement over some widely-used known techniques such as secondharmonic FROG, which cannot differentiate a pulse from its time-reversedreplica, and hence requires additional information regarding the pulseto lift this ambiguity.

In certain embodiments, the dummy pulse 224 is generally not recoveredwell, which is of course inconsequential. FIG. 11 illustrates thisbehavior by showing the original magnitude of the dummy pulse used inthe recovery of FIG. 2 (solid line, reproduced from FIG. 1A) and therecovered magnitude of the dummy pulse (dashed line). Most of thesignificant features of the original dummy temporal profile, such as theoscillations in the tail, are washed out in the recovered dummy pulsetemporal profile. This behavior is expected in certain embodiments inwhich the pulse sequence approximates an MPF, but does not equal a trueMPF. Consequently, certain embodiments described herein which convergesto a true MPF produce a recovered pulse sequence that differs from theoriginal pulse sequence. As shown by FIG. 11, almost all of thedifferences between the recovered pulse sequence and the original pulsesequence occur around the first dominant peak (i.e., the dummy pulse224). This feature is due to the minimum phase condition being strictlyrelated to the dummy pulse's shape and magnitude. Specifically, for afunction to be a true MPF, the rise time of its dominant peak at theorigin (e.g., t=0) is very sharp. Since this condition cannot be metperfectly with practical dummy pulses in the laboratory, the constructedpulse sequence only approximates a true MPF and is not a true MPF. Onthe other hand, the shape of the sample pulse 202 has little bearing onwhether the pulse sequence is a true MPF. In a simplistic view, theinconsequential recovery of the dummy pulse is sacrificed to achieve avery accurate recovery for the weaker sample pulse or pulses.

In certain embodiments, the parameter that influences the accuracy ofthe recovery most strongly is the magnitude of the dummy pulse 224 ascompared to the magnitude of the sample pulse 202. For the recoveryresults shown in FIG. 2, the ratio of the dummy pulse magnitude to thesample pulse magnitude (referred to herein as the magnitude ratio) waschosen to equal 40. FIG. 12 is a plot of the logarithm of the errors inthe recovered magnitude and phase (as compared to the original magnitudeand phase) for the optical pulses of FIGS. 9A and 9B as a function ofthe logarithm of the magnitude ratio. The error plotted in FIG. 12 wasdefined as$\frac{\int{{{{f(t)} - {\hat{f}(t)}}}^{2}{\mathbb{d}t}}}{\int{{{f(t)}}^{2}{\mathbb{d}t}}},$where f(t) and f(t) are the original and the recovered quantities,respectively, and where the integrals were calculated over the timeduration of the sample pulse only. FIG. 12 demonstrates that forembodiments with a magnitude ratio of approximately 6 or greater, theerror in both the recovered phase and the recovered magnitude dropsdramatically as compared to embodiments with a magnitude ratio less thanapproximately 6. Above a magnitude ratio of approximately 6, the erroris roughly constant and less than approximately 2×10⁻⁴. The magnituderatio at which the large reduction of the error is observed (e.g., thecritical ratio) depends on the functional form of both the sample pulseand the dummy pulse, and it typically ranges between approximately 5 andapproximately 15. In certain embodiments, the dummy pulse is selected toprovide a magnitude ratio greater than approximately 20, while in otherembodiments, the dummy pulse is selected to provide a magnitude ratiogreater than approximately 30. In certain embodiments, the convergenceof the method can be checked by carrying out the spectral measurementfor two values of the magnitude ratio (e.g., 25 and 50), and determiningthat the difference between the two solutions is negligible. FIG. 12also illustrates that the recovery remains equally good with magnituderatios as large as 1000. Such large magnitude ratios can be used in thedetermination of ultra-weak optical pulses. Certain embodimentsdescribed herein characterize a temporal profile of an ultrashort pulsethat is at least 1 million times weaker in peak power than the leadingdummy pulse.

In certain embodiments, errors and noise in the measured power spectrumaffect the accuracy of the recovered temporal profiles. A simulatednoise-free measured original power spectrum, plotted in FIG. 13 (dashedcurve), was calculated by taking the square of the theoretical FTmagnitude of an arbitrary dummy-sample pulse sequence with a peakmagnitude ratio of 13. A simulated noisy measured power spectrum,plotted in FIG. 13 (solid curve), was calculated by multiplying thenoise-free power spectrum by a uniform random noise with a 20%peak-to-peak amplitude and an average of unity. FIGS. 14A and 14B areplots of the magnitude and phase, respectively, of the two recoveredtemporal profiles obtained by applying the SIMBA method to each of thespectra of FIG. 13. The solid lines of FIGS. 14A and 14B correspond tothe recovered temporal profile corresponding to the noise-free powerspectrum of FIG. 13, and the dashed lines of FIGS. 14A and 14Bcorrespond to the recovered temporal profile corresponding to the noisypower spectrum of FIG. 13. The recovery is still quite good in spite ofthe large measurement noise, with the mean error in the recovered pulseintensity less than 1.5%. Simulations also show that the mean error inthe recovered temporal profile is proportional to the error in the powerspectrum. Such results illustrate that certain embodiments describedherein work well even with fairly noisy power spectrum measurements. Thenoise sensitivity is affected by the magnitude ratio of the dummy pulseto the sample pulse. In certain embodiments in which the main source ofnoise in the OSA measurement system is proportional to the input power,as assumed above, an increase in the magnitude of the dummy pulseresults in a larger amount of noise in the measured power spectrum, anda larger error in the recovered sample pulse temporal profile. Tomaximize the accuracy of the recovered temporal profile in the presenceof noisy power spectrum measurements, certain embodiments advantageouslyselect a magnitude ratio close to the critical ratio (e.g., in a rangebetween approximately 5 and approximately 15). For example, themagnitude ratio corresponding to FIG. 13 was selected to be 13, whichadvantageously facilitates both accurate convergence of the iterativemethod and reduced sensitivity to measurement noise.

The accuracy of certain embodiments described herein is also affected bythe frequency bandwidths of the dummy pulse and the sample pulse. Incertain embodiments, the frequency bandwidth of a pulse is defined to bethe full width of the FT spectrum magnitude at 10% of its maximum value.In the numerical example illustrated by FIG. 10, the frequency bandwidthof the dummy pulse is approximately 4.5 times that of the sample pulse(i.e., a frequency bandwidth ratio of approximately 4.5). This can beseen as well in the power spectrum of FIG. 1B, where the narrow band inwhich the interference fringes occur represents roughly the frequencyrange of the narrower sample pulse.

FIGS. 15A and 15B are plots of the sample pulse and the dummy pulse,respectively, with a frequency bandwidth ratio of approximately 2. Thesolid lines of FIGS. 15A and 15B correspond to the pulse magnitudes, andthe dashed lines of Figures 15A and 15B correspond to the pulse phases.In this numerical example, the temporal width of the dummy pulse ischosen to be approximately 2.4 times more narrow than the temporal widthof the sample pulse, as illustrated by FIGS. 15A and 15B.

FIGS. 16A and 16B are plots of the FT spectra of the sample pulse andthe dummy pulse, respectively, of FIGS. 15A and 15B, with solid linescorresponding to FT magnitudes and dashed lines corresponding to FTphases. The FT spectra of FIGS. 16A and 16B include strong third-orderand second-order phases, respectively. In contrast, in the exampleillustrated by FIGS. 10A and 10B, the sample pulse of FIG. 10A and thedummy pulse of FIG. 10B included strong second-order and third-orderphases, respectively. As a result, the temporal profiles of the twopulses of FIGS. 15A and 15B are quite different from those of FIGS. 9Aand 9B discussed above. As used herein, the term “second-order phase”refers to the term of the phase that is proportional to the square ofthe normalized optical frequency and the term “third-order phase” refersto the term of the phase that is proportional to the cube of thenormalized optical frequency.

FIG. 17A illustrates the electric field of a pulse sequence formed bydelaying the sample and dummy pulses of FIGS. 15A and 15B. In the pulsesequence of FIG. 17A, the magnitude ratio of the two pulses was chosento be 30. The computed square of the FT magnitude of this pulse sequenceis shown in FIG. 17B. The sample pulse recovered by applying anembodiment of the SIMBA method to FIG. 17B, shown by the dashed lines ofFIG. 18, is again in very good agreement with the original pulse, shownby the solid lines of FIG. 18. This result demonstrates that a frequencybandwidth ratio of approximately 2 is still high enough for a reliablerecovery.

However, reducing the frequency bandwidth ratio much further (e.g., toless than 1) would introduce a noticeable error in the recoveredtemporal profile. This behavior is explained by observing that if thefrequency bandwidth of the dummy pulse is narrower than the frequencybandwidth of the sample pulse, then some of the high frequencycomponents in the FT magnitude spectrum (e.g., see FIGS. 1B and 17B)will be missing. Since these high-frequency oscillations carryinformation regarding the sample pulse, the sample pulse will not befaithfully recovered from the FT magnitude spectrum. For the recovery tobe accurate, a minimum frequency bandwidth ratio of greater than 1(e.g., 2 or more) is advantageously used in certain embodiments.

The minimum frequency bandwidth selection is not specific to certainembodiments of the method described herein. Most other SI techniquesalso utilize a reference pulse with a broader frequency spectrum thanthe ultrashort pulses to be characterized. Note that with certainembodiments described herein, there is no maximum frequency bandwidthratio requirement. In certain embodiments, the dummy pulse frequencybandwidth can be as much as 1000 times wider than the sample pulsefrequency bandwidth. In practice, the maximum dummy pulse bandwidth willbe imposed by the available laser. In certain embodiments, a dummy pulsewith a sufficient frequency bandwidth (e.g., about twice the frequencybandwidth of the sample pulse) can be easily obtained by compressing alonger pulse (e.g., the sample pulse itself) using one of many pulsecompression techniques available in the prior art (see, e.g., M. Nisoliet al., “Generation of high energy 10 fs pulses by a new pulsecompression technique,” Appl. Phys. Lett., Vol. 68, pp. 2793-2795(1996); M. A. Arbore et al., “Engineerable compression of ultrashortpulses by use of second-harmonic generation in chirped-period-poledlithium niobate,” Opt. Lett., Vol. 22, pp. 1341-1342 (1997)).

As shown in FIG. 18, there is a slight dc offset between the recoveredand original phase spectra. This offset corresponds physically to therelative phase of the electric-field oscillations under the complexelectric-field envelope e(t). For many applications, this offset isinconsequential and previously-existing techniques also cannot recoverthis dc phase component.

Certain embodiments described herein use a single power spectrummeasurement to advantageously characterize the complex electric fieldprofile of a series of sample laser pulses (as might be generated forexample by multiple laser sources). FIG. 19A illustrates an exemplaryseries of a single dummy pulse and two sample pulses formed by delayingtwo different ultrashort laser pulses. The single dummy pulse in FIG.19A is chosen to be the same as that in FIG. 9B. The relative maximumelectric field magnitudes of the dummy pulse, the first sample pulse,and the second sample pulse of FIG. 19A were chosen to be 20, 1 and ⅓,respectively. The recovery results were independent of these relativemagnitudes. For example, relative magnitudes of 40, 1, 1 and 30, 1, ½,gave similar results. FIG. 19B illustrates the computed square of the FTmagnitude of the complex temporal profile of FIG. 19A. Applying the sameiterative error-reduction method as before to the measured powerspectrum of FIG. 19B simultaneously recovers the complex electric fieldtemporal profile of both sample pulses, as shown by the dashed curves ofFIG. 20. The recovery is as accurate as in the previous examplesdescribed above. Thus, certain embodiments advantageously recoversimultaneously the full complex electric field temporal profiles of twodifferent ultrashort pulses using a single power spectrum measurement.Equally accurate recoveries are obtained when the two sample pulses havedifferent temporal profiles. There is again a small inconsequential dcoffset in the recovered phase spectrum. In the time interval between thetwo sample pulses, the error in the recovered phase spectrum is simplydue to the fact that the magnitude of the electric field in thatinterval is very close or equal to zero, as discussed earlier.

Certain embodiments described herein can advantageously characterizepulse sequences containing many more than two individual sample pulses.However, when the number of sample pulses is too large, the oscillationsin the FT magnitude arising from multiple interference between thesample pulses becomes so rapid that a higher resolution OSA is used tomeasure the power spectrum. Therefore, the number of sample pulses thatcan be simultaneously characterized depends on the resolution of theOSA.

In certain embodiments, various ultrashort pulse shaping techniques(see, e.g., M. M. Wefers and K. A. Nelson, “Analysis of programmableultrashort waveform generation using liquid-crystal spatial lightmodulators,” J. Opt. Soc. Am. B, Vol. 12, pp. 1343-1362 (1995); A.Rundquist et al., “Pulse shaping with the Gerchberg-Saxton algorithm,”J. Opt. Soc. Am. B, Vol. 19, pp. 2468-2478 (2002)) can be used to modifythe temporal profile of the dummy pulse in order to achieve anabsolutely true MPF for the pulse sequence's electric field. Certainsuch embodiments can potentially have a dramatically improved recoveryspeed. In principle, by using a true MPF, certain embodiments describedherein can converge in less than 5 iterations, thus cutting down thecomputation time to a fraction of a second, even when using a relativelyslow programming environment such as MATLAB 5.

FIG. 21 schematically illustrates an example measurement configurationcompatible with certain embodiments described herein and used in thefollowing two example measurements. A femtosecond laser (not shown) isused to produce an input pulse 240 to characterize an output samplepulse 202 that results from the interaction of the input pulse 240 withan optical system 250 of interest. A first portion of the intensity ofthe input pulse 240 is transmitted through a mirror 252 and anattenuator 253 to interact with the optical system 250 of interest. Theresultant sample pulse 202 is then reflected by two mirrors 254, 256 andsent to the optical spectrum analyzer 232. A second portion of theintensity of the input pulse 240 is reflected by the mirror 252 and themirror 258 and is transmitted through the mirror 256 and sent to theoptical spectrum analyzer 232 and serves as the dummy pulse 224. Thecombination of the sample pulse 202 and the dummy pulse 224 creates apulse sequence 222.

The mechanical stability of the measurement configuration was much lessthan one micron, which directly means that the delay jitter in themeasurements would be lower than one femtosecond. In addition, themeasurement equipment was kept at room temperature such that themeasurements were far away from being shot-noise limited. However, themeasurement configuration was able to recover the sample pulse complexelectric field profiles quite reliably.

EXAMPLE 1

In the first example, the optical system 250 of interest comprised aslab of fused silica approximately 16 centimeters long. Thus, themeasurement of this first example can be considered to be a materialcharacterization measurement. The input pulse 240 had afull-width-at-half-maximum width of approximately 145 femtoseconds, andwas generated from a Ti:sapphire oscillator that ran at approximately859 nanometers with a repetition rate of approximately 96 MHz. The peakpowers of the dummy pulse 224 and the sample pulse 202 wereapproximately 2.61 microwatts and 168 nanowatts, respectively, whichcorresponds to a maximum field ratio of approximately 4.

FIG. 22 is a graph of the measured power spectrum of the pulse sequence222 formed by delaying the dummy pulse 224 with respect to the samplepulse 202. Because fused silica has a large transparency window withχ⁽²⁾≈0, the bandwidth of the sample pulse 202 roughly matches that ofthe dummy pulse 224. Thus, the interference between the dummy pulse 224and the sample pulse 202, as shown in the measured power spectrum ofFIG. 22, occurs across the available bandwidth. The effect of the thickslab of fused silica is simply to broaden the sample pulse 202 with astrong second-order spectral phase, without affecting the bandwidth ofthe dummy pulse 224.

FIGS. 23A and 23B are graphs of the resulting intensity and phase,respectively, of the recovery after applying the SIMBA techniquedisclosed herein to the square-root of the measured power spectrum shownin FIG. 22. For comparison purposes, FIGS. 23A and 23B also include theresults of the recovery obtained using both the FROG technique and theTADPOLE technique. The general agreement in the recovery results, forboth the intensity and the phase of the electric field profile, obtainedusing the SIMBA, FROG, and TADPOLE techniques is very good. The recoveryresults of the TADPOLE technique, which also relies on the measuredpower spectrum shown in FIG. 22, involves two additional measurements:an initial full characterization of the dummy pulse 224 using a FROGconfiguration, and then an additional power spectrum measurement for theunknown sample pulse 202 alone. In contrast, the SIMBA technique onlyused the measured power spectrum shown in FIG. 22 for the recovery. Inthis example, the temporal FWHM of the dummy pulse 224 is only about 1.7times more narrow than the FWHM of the sample pulse 202, and therecovery results obtained using the SIMBA technique are still quitegood. In particular, the spectral phase, as shown in FIG. 23B, agreesvery well with the phases obtained using the other techniques and withthe predicted phase spectrum that can be theoretically computed usingthe known dispersion coefficients of fused silica. For the spectralphase recovery, after a certain range in which the intensity of thesample pulse 202 drops significantly, the phase curves obtained usingthe various techniques start to diverge from one another. This result isexpected and is inconsequential.

EXAMPLE 2

In this example, the optical system 250 of interest comprises a thinfilm bandpass filter, having a FWHM of approximately 10 nanometers, thatsignificantly filters the frequency bandwidth of the dummy pulse 224.This spectral filtering resulted in a temporally wider sample pulse,while the dummy pulse 224 had a FWHM of about 30 femtoseconds. To testthe repeatability of the SIMBA technique, using the measurementconfiguration of FIG. 21, two successive measurements were made withslightly different delay values between the sample pulse 202 and thedummy pulse 224. FIGS. 24A and 24B show recorded charge-coupled-device(CCD) images at the optical spectrum analyzer 232 for these twosuccessive measurements. The resolution of the optical spectrum analyzerwas about 54 picometers. For the measurement of FIG. 24A, a maximumfield ratio between the dummy pulse 224 and the sample pulse 202 ofabout 4.40 was used and for the measurement of FIG. 24B, a maximum fieldratio between the dummy pulse 224 and the sample pulse 202 of about 4.17was used.

The recorded CCD images of FIGS. 24A and 24B are two-dimensional. Sincethe complex electric field profile of the pulses are one-dimensional, inprinciple, only an array of CCD pixels along a single line (e.g., alongthe x-direction) would be sufficient. However, to improve thesignal-to-noise ratio in the measurement, a two-dimensional CCD arraywas used. FIGS. 25A and 25B are plots of the measured power spectrumcorresponding to the input pulse sequence obtained from FIGS. 24A and24B, respectively, by adding the recorded spatial Fourier transformmagnitude spectra along the vertical axis.

A comparison of FIG. 22 with FIGS. 25A and 25B illustrates that thepresence of the bandpass filter in Example 2 significantly reduces thebandwidth of the sample pulse 202. This bandwidth reduction results inthe interference occurring only at the center region of the wholeavailable bandwidth of the dummy pulse 224, as shown in FIGS. 25A and25B.

FIGS. 26A and 26B show the intensity and phase, respectively, of theelectric field of the sample pulse 202 recovered by applying the SIMBAtechnique to the square-root of the measured power spectra of FIGS. 25A(“SIMBA measurement #1”) and 25B (“SIMBA measurement #2”). Forcomparison purposes, FIGS. 26A and 26B also show the results of a FROGtechnique applied to the same sample pulse. The agreement between theresults of the SIMBA technique and the FROG technique are quite good.The FROG technique used to produce the corresponding plots of FIGS. 26Aand 26B was based on second-harmonic generation, and therefore had timereversal ambiguity in its result. This time reversal ambiguity iscorrected by the SIMBA technique, such that FIG. 26A has the correcttime axis. It is also noteworthy that both techniques reliably recoveredthe satellite pulse having a smaller amplitude between about 200femtoseconds to about 400 femtoseconds. The physical origin of thissatellite pulse is the spectral side lobes created by the bandpassfilter used in this example. The observed discrepancy in the recoveredphase spectra, especially for times less than about 200 femtoseconds, issimply due to a significant reduction of the pulse intensity. Theconsistency between the results of the SIMBA technique applied to thetwo successive measurements illustrates the repeatability of the SIMBAtechnique with different delay and maximum field ratios.

Various embodiments of the present invention have been described above.Although this invention has been described with reference to thesespecific embodiments, the descriptions are intended to be illustrativeof the invention and are not intended to be limiting. Variousmodifications and applications may occur to those skilled in the artwithout departing from the true spirit and scope of the invention asdefined in the appended claims.

1. A method of determining a complex electric field temporal profile ofan optical pulse, the method comprising: (a) providing a measuredmagnitude of the Fourier transform of a complex electric field temporalprofile of a pulse sequence comprising the optical pulse and a dummypulse; (b) providing an estimated phase term of the Fourier transform ofthe complex electric field temporal profile of the pulse sequence; (c)multiplying the measured magnitude and the estimated phase term togenerate an estimated Fourier transform of the complex electric fieldtemporal profile of the pulse sequence; (d) calculating an inverseFourier transform of the estimated Fourier transform, wherein theinverse Fourier transform is a function of time; and (e) calculating anestimated complex electric field temporal profile of the pulse sequenceby applying at least one constraint to the inverse Fourier transform. 2.The method of claim 1, further comprising: (f) calculating a Fouriertransform of the estimated complex electric field temporal profile ofthe pulse sequence; and (g) calculating a calculated phase term of theFourier transform of the estimated complex electric field temporalprofile of the pulse sequence.
 3. The method of claim 2, whereincalculating the calculated phase term of the Fourier transform comprisesusing a logarithmic Hilbert transformation of the Fourier transform ofthe estimated complex electric field temporal profile of the pulsesequence.
 4. The method of claim 2, further comprising: (h) using thecalculated phase term of (g) as the estimated phase term of (c) andrepeating (c)-(e).
 5. The method of claim 4, wherein (c)-(h) areiteratively repeated until the estimated complex electric field temporalprofile of the pulse sequence reaches convergence.
 6. The method ofclaim 5, wherein convergence is reached when a difference betweenestimated complex electric field temporal profiles obtained after twoconsecutive iterations is less than a predetermined value.
 7. The methodof claim 6, wherein the predetermined value is 0.1% of the estimatedcomplex electric field temporal profile of an iteration.
 8. The methodof claim 4, wherein (c)-(h) are iteratively repeated a predeterminednumber of times.
 9. The method of claim 1, wherein providing themeasured magnitude of the Fourier transform of the complex electricfield temporal profile of the pulse sequence comprises: generating apulse sequence comprising the optical pulse and the dummy pulse;measuring a power spectrum of the pulse sequence; and calculating thesquare root of the measured power spectrum.
 10. The method of claim 9,wherein the dummy pulse has a magnitude larger than the magnitude of theoptical pulse.
 11. The method of claim 9, wherein the optical pulse andthe dummy pulse do not substantially overlap one another.
 12. The methodof claim 9, wherein the dummy pulse precedes the optical pulse such thatthe complex electric field temporal profile of the pulse sequence equalsor approximates a minimum-phase function.
 13. The method of claim 9,wherein the optical pulse precedes the dummy pulse such that the complexelectric field temporal profile of the pulse sequence equals orapproximates a maximum-phase function.
 14. The method of claim 1,wherein providing the measured magnitude of the Fourier transform of thecomplex electric field temporal profile of the pulse sequence comprisesproviding a calculated square root of a previously-measured powerspectrum of a pulse sequence comprising the optical pulse and the dummypulse, wherein the optical pulse followed the dummy pulse.
 15. Themethod of claim 1, wherein providing the estimated phase term of theFourier transform of the complex electric field temporal profile of thepulse sequence comprises providing an initial estimated phase term equalto a real or complex constant.
 16. The method of claim 15, wherein theinitial estimated phase term is a previously-stored function retrievedfrom a computer system.
 17. The method of claim 15, wherein the initialestimated phase term is a phase calculated from a previous opticalpulse.
 18. The method of claim 15, wherein the initial estimated phaseterm is calculated from the measured magnitude using a logarithmicHilbert transform.
 19. The method of claim 1, wherein the complexelectric field temporal profile of the pulse sequence equals orapproximates a minimum-phase function and applying the at least oneconstraint to the inverse Fourier transform comprises setting theinverse Fourier transform to zero for times less than zero.
 20. Themethod of claim 1, wherein the complex electric field temporal profileof the pulse sequence equals or approximates a maximum-phase functionand applying the at least one constraint to the inverse Fouriertransform comprises setting the inverse Fourier transform to zero fortimes greater than zero.
 21. The method of claim 1, wherein the complexelectric field temporal response of the pulse sequence has a knowntemporal duration and applying the at least one constraint to theinverse Fourier transform comprises setting the inverse Fouriertransform to zero for times greater than the known temporal duration ofthe complex electric field temporal profile of the pulse sequence. 22.The method of claim 1, wherein the optical pulse has a predeterminedmaximum peak power and applying the at least one constraint to theinverse Fourier transform comprises constraining the magnitude of theinverse Fourier transform to be less than or equal to the maximum peakpower.
 23. The method of claim 1, wherein the pulse sequence furthercomprises a second optical pulse, and the method further comprisesdetermining a complex electric field temporal profile of the secondoptical pulse.
 24. The method of claim 23, wherein the pulse sequencecomprises a single dummy pulse.
 25. A computer-readable medium havinginstructions stored thereon which cause a general-purpose computer toperform the method of claim
 1. 26. A computer system comprising: meansfor estimating an estimated phase term of a Fourier transform of acomplex electric field temporal profile of a pulse sequence comprisingan optical pulse and a dummy pulse; means for multiplying a measuredmagnitude of the Fourier transform of the complex electric fieldtemporal profile of the pulse sequence and the estimated phase term togenerate an estimated Fourier transform of the complex electric fieldtemporal profile of the pulse sequence; means for calculating an inverseFourier transform of the estimated Fourier transform, wherein theinverse Fourier transform is a function of time; and means forcalculating an estimated complex electric field temporal profile byapplying at least one constraint to the inverse Fourier transform.